Journal of Process Control 19 (2009) 539–549

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Journal of Process Control

journal homepage: www.elsevier .com/locate / jprocont

Application of reduced models for robust control and state estimationof a distributed parameter system

Limei Ding *, Andreas Johansson, Thomas GustafssonDivision of Systems and Interaction, Luleå University of Technology, SE-97187 Luleå, Sweden

a r t i c l e i n f o a b s t r a c t

Article history:Received 21 March 2007Received in revised form 24 April 2008Accepted 25 April 2008

Keywords:Distributed parameter systemReduced modelsModel uncertaintyPaper pulp digesterRobust controlState estimation

0959-1524/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.jprocont.2008.04.009

* Corresponding author. Tel.: +46 70 3581716.E-mail address: limei.ding@ltu.se (L. Ding).

This paper studies the application of reduced models of a distributed parameter system for robust processcontrol and state estimation. We take the approach of integrating model reduction, parameter identifica-tion, and model uncertainty analysis, in purpose to find an appropriate trade-off between complexity androbust performance. The application example is the temperature system in a continuous paper pulpdigester. Physical modeling of this process results in coupled linearized partial differential equationswhich are then reduced into low-order nominal process models using an orthogonal collocation approx-imation method.

Two different approaches to obtaining a model uncertainty description are adapted for use on a dis-tributed parameter system with low-order nominal model and shown to produce similar results whentested with measurement data. It is also demonstrated how this uncertainty description, in combinationwith the reduced model, may be used for robust control design and verification of the control perfor-mance on the distributed parameter system.

Finally, the possibility of estimating the distributed process state using a state observer for thereduced process is demonstrated. Measurements of the process state in a certain position is availableand is shown to agree with the estimated state at the same position.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

The control of distributed parameter systems (DPS) is an impor-tant issue in a wide range of the applications, such as spatial pro-files, size distributions, fluid flows and material microstructure [1].First principles modeling of distributed processes lead to nonlinearDPSs of many forms [2], e.g. hyperbolic and parabolic partial differ-ential equations, Navier–Stokes equations, and integro-differentialequations. This raises difficulties when it comes to controller syn-thesis and implementation due to the infinite dimensionality.

In practice, a model approximation is often required in order toproduce a low-order model that captures the dominant character-istics of the system [3,4]. The topic of model order reduction hasbeen actively discussed in the research field of DPS control. In[5], Li and Christofides applied different model order reduction ap-proaches to a diffusion–convection–reaction process, such as thefinite difference method, the Galerkin projection approach, andthe orthogonal collocation on finite elements. The reduced modelswere used in process simulation and control design. Lee et al. [6]presented a general algorithm for robust model reduction of a non-linear PDE. The study is based on five model order reduction tech-niques and the authors proposed a resulting model with the lowest

ll rights reserved.

modeling error in the frequency range important for controldesign.

Also, many first principle models contain unknown parameters,therefore an identification approach for the parameter estimationneeds to be considered [7–9]. A low-order model approximationwith identified parameters provides a viable solution for obtaininga nominal model suitable for model-based control and monitoringof an industrial DPS. This approach is a grey-box one, because itcombines the first principle modeling with data driven parameteridentification.

In order to achieve robust stability and performance for theclosed-loop, it is necessary to deliver not only a nominal model,but also a reliable estimate of the uncertainty associated withthe model. This is one of the main objectives of control-orientedidentification which was defined by Helmicki et al. [10] in 1991.Three requirements were set up for control-oriented system iden-tification through the combined use of a system identificationmethod and a robust control design method; (i) the system identi-fication method asymptotically reduces the level of plant uncer-tainty to the point where a robust controller exists; (ii) thesystem identification method yields, at each step, an explicit boundon the remaining plant uncertainty associated with the identifiednominal model; and (iii) this bound must be in a form compatiblewith a robust control design method. Many developments andapplications of control-oriented identification have been discussed

540 L. Ding et al. / Journal of Process Control 19 (2009) 539–549

since then, e.g. a method that is suitable for linear parameter vary-ing systems [11] and an approach that combines physical modelingand system identification [12].

Control of DPSs is an important application area for control-ori-ented identification. Model uncertainty cannot be avoided in thereduction of a DPS, thus the robustness must be considered. In2001, Christofides [13] summarized the methods and applicationsof robust control to a class of PDE systems.

In this paper, we take the approach of integrating model reduc-tion with parameter identification and model uncertainty analysis,in purpose to find an appropriate trade-off between complexityand robust performance. In particular, the convergence of the iden-tified model parameters with increased model order is analyzed.The model error model approach introduced by Ljung and his co-workers [14,15] is applied to evaluate the uncertainty associatedwith the nominal model. Another approach to uncertainty model-ing is also applied where the nominal process model is comparedto the physical process model with parameter values taken froma set of identified values. It is found that the two approaches agreein the most important frequency range.

The low-order nominal process models are then tested in stateestimation and robust control. The H1 loop shaping approach[16,17] makes a trade-off between performance and robustnessby synthesizing an H1 optimal controller with an appropriaterobustness margin and a desired open-loop shape. A Kalman filteris suggested to generate process state estimates for the purpose ofprocess monitoring. Although the Kalman filter is built using a low-order model, it is able to estimate the infinite dimensional state ofthe DPS.

One of the subprocesses studied in [18], a temperature processof the co-current zone in a continuous paper pulp digester at theM-real [19] pulp mill in Husum, Sweden is used as an applicationexample. In the vital steps of the work, i.e. parameter identificationand model uncertainty analysis, we use measurement data fromthis plant. Measurement data are also used for validating the stateestimates obtained with the reduced models. For verifying controlperformance, a frequency domain solution of the original PDE isused to simulate the process.

The outline of the paper is given below: Next section introducesa simplified linear physical model of a process with transport phe-nomena. Both an infinite dimensional model and finite dimen-sional approximations of the model are discussed, thenparameter identification of this process is studied. In Section 3,the errors of the reduced models are estimated and upper boundsof the errors are defined. The basic principle of loop-shaping isbriefly presented in Section 4, then robust control and state esti-mation using the reduced models are demonstrated. Finally, the re-search work is concluded in Section 5.

Fig. 1. Finite set of the measurements and actuators in vertical reactors.

2. A physical model and the model approximation

Conservation laws of mass, momentum and energy form the ba-sis of the field of transport phenomena. These laws applied to theflow of fluids result in the partial differential equations which de-scribe the dynamics of variables of a system, such as temperaturesand concentrations, with respect to time and position. A verticaltubular reactor is an example of such processes with transportphenomena.

2.1. Control of a DPS and its implementation problem

A vertical tubular reactor of co-current flow in two phases, li-quid phase and solid phase, is a DPS and can be modeled by PDEsbased on conservation laws. A direct analytical solution of the con-trol synthesis problem for a DPS modeled by PDEs often results in a

controller that requires distributed measurements and actuators.In practice such a controller is not implementable. On the left sideof Fig. 1, a vertical tubular reactor in common uses is illustrated.There are a finite set of the measurements and actuators, two in-puts and two outputs. Satisfactory control of the process must beachieved using this finite set of measurements and actuators. A fi-nite dimensional control synthesis can be made based on a reducedprocess model, which describes the process dynamics at a finite setof positions, the boundary points z0 and zN+1, and N interior pointsz1,z2, . . . ,zN.

On the right side of Fig. 1, a specified process is shown and theprocess has the main features of a vertical reactor. This is the co-current zone in a continuous paper pulp digester and a detailedintroduction of the digester can be found in [18]. The top positionof this process is denoted as z0, the upper extraction at the bottomposition is denoted as zext and the positions of the three measure-ments are z0, zc6 and zext. The dynamics of the flow temperatures inthe co-current zone of a digester can be modeled using a simplifiedphysical model consisting of coupled PDEs [20,18]:

oT fðz; tÞot

¼ �vfoT f ðz; tÞ

ozþ dfðTcðz; tÞ � T fðz; tÞÞ ð1aÞ

oTcðz; tÞot

¼ �vcoTcðz; tÞ

ozþ dcðT f ðz; tÞ � Tcðz; tÞÞ ð1bÞ

where Tf(z, t) and Tc(z, t) are the temperatures of the liquid phaseand the solid phase, for the specified process in a digester, free li-quor and wood chips; the model parameters vf and vc deal withthe velocities of the liquid flow and solid flow; and the two addi-tional model parameters df and dc are related to the interphase dif-fusion of the two flows. The independent variables z and t are thevertical position of the reactor and the time, respectively. The ver-tical length is normalized, so we define that the top position isz0 = 0 and the bottom position is zext = 1.

The steam temperature at the top of the digester is selected as amanipulated variable, and the temperatures of the both flows atthe top position are assumed to be the same as the steam temper-ature, i.e. Tf(z0, t) = Tc(z0, t) = Ts(z0, t). The temperature at the upperextraction is chosen as the output of the process. Thus the inputand the output are defined as

uðtÞ ¼ Tsðz0; tÞ; yðtÞ ¼ T f ðzext; tÞ ð2Þ

2.2. Model solution in frequency domain

Eqs. (1a) and (1b) formulate a time domain model of the pro-cess. The frequency domain model, i.e. the process transfer func-tions, can be found directly from (1a) and (1b), and is given as

L. Ding et al. / Journal of Process Control 19 (2009) 539–549 541

T f ðz; sÞ ¼ G1ðz; sÞTsðsÞ ð3ÞTcðz; sÞ ¼ G2ðz; sÞTsðsÞ ð4Þ

where the infinite dimensional transfer functions G1(z,s) and G2(z, s)are

G1ðz; sÞ ¼b1 � ðk2ðsÞ þ a1ðsÞÞ

k1ðsÞ � k2ðsÞek1ðsÞz

þ ðk1ðsÞ þ a1ðsÞÞ � b1

k1ðsÞ � k2ðsÞek2ðsÞz ð5Þ

G2ðz; sÞ ¼b2 � ðk2ðsÞ þ a2ðsÞÞ

k1ðsÞ � k2ðsÞek1ðsÞz

þ ðk1ðsÞ þ a2ðsÞÞ � b2

k1ðsÞ � k2ðsÞek2ðsÞz ð6Þ

where

a1ðsÞ ¼sþ df

vf; b1 ¼

df

vfð7Þ

a2ðsÞ ¼sþ dc

vc; b2 ¼

dc

vcð8Þ

and

k1;2ðsÞ ¼12�ða1ðsÞ þ a2ðsÞÞð Þ

� 12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða1ðsÞ þ a2ðsÞÞ2 � 4ða1ðsÞa2ðsÞ � b1b2Þ

qð9Þ

The derivation of the transfer function can be referred to [20].

2.3. Model reduction and model parameter estimation

To deal with the infinite dimensionality of the process, a modelreduction method must be applied, by which an infinite dimen-sional model can be reduced to a low-order (finite dimensional)approximation.

0 5 10 15 20

0.01

0.015

0.02

Number of collocation points

Est

imat

ed v

alue

s of

vf

0 5 10 15 20–2

0

2

4

6

8

10

12

14x 10

–3

Number of collocation points

Est

imat

ed v

alue

s of

vc

Fig. 2. Unknown parameter estimates as a func

In our previous research [18], the OC approximation of the PDEs(1a) and (1b) is provided. The reduced state space model can be ex-pressed in the form

_xðtÞ ¼ AxðtÞ þ BuðtÞyðtÞ ¼ CxðtÞ þ DuðtÞ

ð10Þ

The state vector of the low-order model is x ¼ ½xTf xT

c �T where

xTf ðtÞ ¼ ½T fðz1; tÞ T fðz2; tÞ � � � T f ðzNþ1; tÞ�

xTc ðtÞ ¼ ½Tcðz1; tÞ Tcðz2; tÞ � � � TcðzNþ1; tÞ�

i.e. the temperatures in the collocation points z = z0,z1, . . . ,zN+1, andN is defined as the number of interior collocation points. The matri-ces A, B, C and D are

A ¼�vf A1 � df I df I

dcI �vcA1 � dcI

� �

B ¼�vf B1

�vcB1

� �C ¼ C1 0½ �D ¼ 0

ð11Þ

and

A1 ¼

ol1oz jz¼z1

� � � olNþ1oz jz¼z1

..

. ... ..

.

ol1oz jz¼zNþ1

� � � olNþ1oz jz¼zNþ1

26664

37775 ð12Þ

B1 ¼ ol0oz jz¼z1

� � � ol0oz jz¼zNþ1

h iTð13Þ

C1 ¼ l1ðzNþ1Þ � � � lNþ1ðzNþ1Þ½ � ð14Þ

where the functions lj(z) are Lagrange interpolation polynomialswith the interpolation points chosen as the interior collocationpoints z1,z2, . . . ,zN, and the boundary points z0 = 0 and zN+1 = 1.

0 5 10 15 20–0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Number of collocation points

Est

imat

ed v

alue

s of

df

0 5 10 15 20–0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Number of collocation points

Est

imat

ed v

alue

s of

dc

tion of the number of collocation points N.

Table 1Parameter intervals

Parameter Interval

vf Vf = [0.0162,0.0170]df Df = [0.0659,0.0880]vc Vc = [0.0000, 0.0003]dc Dc = [0.1748,0.2040]

10–3 10–2 10–1 100 10110–2

10–1

100

Am

plitu

de

10–3 10–2 10–1 100 10110–2

10–1

100

Frequency (rad/min)

Am

plitu

de

Fig. 3. Upper: PDE model (3) and (4) with parameter values from Table 1 (grey,dashed) and nominal model (black, solid). Lower: Difference between PDE modeland nominal model (grey, dashed) with upper bound lA (black, solid) and selecteduncertainty weight WA1 (black, dot-dashed).

542 L. Ding et al. / Journal of Process Control 19 (2009) 539–549

The identity matrix ‘‘I” and the zero matrix ‘‘0” that appear in thematrices A and C have the same dimensions as matrices A1 andC1, respectively. For details of the state space model, please referto [18].

For an arbitrary position z, the following approximation is avail-able [18]:

T f ðz; tÞ ¼XNþ1

j¼1

T fðzj; tÞljðzÞ þ TsðtÞl0ðzÞ ð15Þ

Tcðz; tÞ ¼XNþ1

j¼1

Tcðzj; tÞljðzÞ þ TsðtÞl0ðzÞ ð16Þ

The parameter vector of the reduced model is

h ¼ vf df vc dc½ �T ð17Þ

and it is the same as that can be defined for the infinite dimensionalphysical model (1a) and (1b). In [18] the four unknown parametersshown in (17) are estimated using the least squares method. The re-duced models with N = 3, 5, 7, 9 are selected for the parameter iden-tification, and the data used for both parameter identification andmodel validation are measurement data from the pulp digester atthe M-real paper plant in Husum, Sweden.

In the current research we extend the identification work byselecting more of the reduced models with different choice of N.It appears that the parameter values converge to small intervalswhen the number of collocation points N increases. Fig. 2 showsthe parameter values as functions of N for a number of identifica-tion experiments. It can be seen from Fig. 2 that for N P 11, theparameter values are confined to small intervals, and those inter-vals are specified in Table 1. The OC approximations with differentselection of N provide a set of nominal models for the model appli-cations studied later in this paper.

3. Model error analysis

We assume that the ‘‘true” process belongs to the set P of pro-cesses Gp, where Gp denotes a perturbed process. Robustness of acontroller then implies that it satisfies its specification for allGp 2P. Process perturbations can be constructed in different ways[17], such as additive uncertainty, multiplicative uncertainty, nor-malized-coprime-factor uncertainty, etc. Choosing the additiveuncertainty for our discussion, P consists of all Gp on the form

GpðsÞ ¼ bGðsÞ þWAðsÞDAðsÞ ð18Þ

where bGðsÞ is a nominal process model, WA(s) is an uncertaintyweight, and DA(s) is an arbitrary complex function satisfyingjDA(ix)j < 1 for all frequencies, and the subscript A in WA and DA de-notes additive uncertainty.

In the preceding section, a set of nominal models were obtainedby taking the OC approximations of the PDEs (1a) and (1b), andusing the parameter estimates shown in Fig. 2. A satisfactory ro-bust controller synthesized from such a nominal model must beable to tolerate the model perturbations due to model simplifica-tion, model order reduction and the parameter estimation. Thusit is interesting to analyze the perturbations in the frequencydomain.

3.1. Model errors caused by parameter perturbation and model orderreduction

One may separately discuss the model perturbations due to dif-ferent reasons. In this subsection we discuss the model errorscaused by parameter perturbation and model order reduction.We assume that a subset P1 �P consists of the PDE transfer func-tion (3) and (4) with different parameter values. Using the param-eter value intervals specified in Table 1, an upper bound lA(w) forthe magnitude jWA(ix)j of the uncertainty weight can then be ob-tained as [17]:

lAðxÞ ¼maxh2HjG1ð1; ixÞ � bGðixÞj ð19Þ

where h = (vf,df,vc,dc) and H = Vf � Df � Vc � Dc is the Cartesianproduct of the intervals in Table 1.

As nominal model bGðsÞ, the identified low-order model (11)with N = 7 and parameter values vf = 0.0161, df = 0.0476,vc = 0.0078, dc = 0.0420 is selected. Note that these parameter val-ues are valid only for this particular low-order model. They do notbelong to the parameter intervals of Table 1 since they are not nec-essarily related to the true parameter values of the process.

The upper plot in Fig. 3 shows the amplitude of the frequencyresponse of the nominal model and the PDE model (3) and (4) withparameter values selected from the intervals in Table 1 while thelower plot shows the amplitude of the differencejG1ð1; ixÞ � bGðixÞj. The final step in selecting the uncertaintyweight is to find a simple low-order weight WA1(s) that satisfiesjWA1(ix)jP lA(x) for all x P 0. Our choice

WA1ðsÞ ¼ 12sðs=2þ 1Þðs=0:06þ 1Þ2 ð20Þ

is shown in the figure with a dot-dashed line.Because lA(x) defined in (19) is obtained by taking the differ-

ence between the PDE model that is perturbed by the uncertainparameters and the reduced model, it is clear that the weight func-tion (20) gives an upper bound of the model error due to theparameter perturbation and the model order reduction. However,this error bound does not include the model errors caused by mod-el simplification, etc. The weight function WA1 thus only partly rep-resents the model errors of a reduced OC model.

10–3

10–2

10–1

100

10–3

10–2

10–1

100

101

Frequency (rad/min)

Am

plitu

de

Fig. 5. The model error model estimated for the OC model with N = 7.

L. Ding et al. / Journal of Process Control 19 (2009) 539–549 543

3.2. Overall model errors

To analyze the overall errors of a reduced model, one shouldexamine the difference between the simulation data using the re-duced model and measurement data collected from the process,since this difference represents the model errors caused by all dif-ferent reasons.

An approach proposed by Ljung [14] that deals with model errormodels is an alternative for the model error estimation. We expressa real plant as follows:

y ¼ Guþ v ð21Þ

where y is the observed output variable of the plant, G is the trueprocess and v is the system noise. A model residual e can becomputed

e ¼ y� bGu ð22Þ

where bG is a nominal model that is estimated from the data y and u.The model residual e can be separated into model error D and dis-turbances w [14]:

e ¼ Duþw ð23Þ

where no further requirement is put on w.Fig. 4 shows the basic idea of model error modeling [14,15]. The

upper plot illustrates the nominal model bG along with the uncer-tainty region (dashed lines) and the center of the uncertainty re-gion, bG þ D. The lower plot demonstrates the model error modelD and its uncertainty band (dashed lines). This uncertainty con-tains zero except for an interval at high frequencies, where thenominal model is outside the so constructed uncertainty regionshown in the upper plot.

A nominal model could fail in a traditional correlation test, butpass a control-oriented model validation test if the norms of itsmodel error model D and disturbance w are limited in an accept-able level [14]. An advantage with the model error model conceptis that the nominal model still can be used for control design, evenif it is falsified by D. This is the case, for example, if the falsificationtakes place in ‘‘unimportant” frequency regions [15].

A black-box identification procedure named prediction errormethod [21,22] is employed to estimate the model error modelsof the nominal models. The model error model of the OC nominalmodel with N = 7 is estimated and the estimate passes the correla-tion test. In Fig. 5, the estimated model error model is plotted usinga solid line, and its 99% confidence interval is plotted using dottedlines.

Fig. 4. The basic idea of model error modeling.

Putting the representations of e given by (22) and (23) together,we have

y ¼ bGuþ Duþw ð24Þ

This equation suggests an additive uncertainty D, and bG þ D repre-sents a perturbed process. We adopt the model error model to rep-resent the perturbed process as follows:

GpðsÞ ¼ bGðsÞ þ DðsÞ ð25Þ

where the model error model D(s) is assumed to be confined withinthe 99% confidence interval. For the control synthesis problem usingthe perturbed model in (25), we assume that the ‘‘true plant” issome Gp(s) 2P.

3.3. Comparison

Ljung et al. [21,15,23] analyzed the error in standard identifica-tion problems. The error can be divided into a bias, due to systemdynamics which is not captured by the estimated nominal model,and a variance, due to noise affecting the data. Although we ob-tained the model in this study by applying a grey-box approach,the main model error still consists of the two parts mentionedabove. The model simplification and the model order reductiondefinitely lead to model structure error, i.e., the bias. The varianceof parameter estimation also exists. Since we use actual measure-ment data in our study, the properties of the disturbances may bemore complicated than that assumed for the standard identifica-tion problems, since there are feedback phenomena involved inthe process due to the liquor circulation, and that the measure-ment noise is not guaranteed to be white. Additional errors mayappear due to these factors.

In Fig. 6, the model error model of the nominal OC model withN = 7 and the weight function defined by (20) are plotted together.We approximately divide the frequency range into three subinter-vals: a low-frequency region (L-region), a mid-frequency region(M-region) and a high-frequency region (H-region). It is obviousthat the estimated model error model is almost covered by WA1

in the M-region, while it exceeds WA1(s) in the L-region and theH-region. Hence the weight function WA1 is an acceptable modelerror description in the M-region.

The upper plots of Fig. 3 demonstrate the frequency propertiesof the infinite dimensional model and a nominal model. The prop-erties in a frequency range approximately corresponding the M-re-gion are characteristic for the process, therefore the M-region is animportant frequency range.

The agreement between the different model error estimates inthe M-region shown in Fig. 6 supports the legitimacy of the simpli-fied physical model (1a) and (1b), and this implies that the errors

10–2

10–1

100

101

Am

plitu

de

10–3 10–2 10–1 10010–3

10–2

10–1

100

101

Frequency (rad/min)

Am

plitu

deM – region

L – region

H – region

Fig. 6. Comparing WA1(s) to the model error model of the reduced OC model withN = 7. Solid line: estimate of the model error model; dotted lines: 99% confidenceinterval of the model error model; dot-dashed line: weight function WA1(s).

544 L. Ding et al. / Journal of Process Control 19 (2009) 539–549

of the reduced models in this frequency region are caused mainlyby the parameter perturbation and the model order reduction.We have estimated model error models for the OC models withN = 3,5, . . . ,17,19, then compared to WA1. The results are similaras those for N = 7.

The model error model can also be used for assessing the prin-cipal deficiencies of the physical model and improve it. Fig. 6shows that the main differences between the model error modeland WA1(s) appear particularly in the L-region and the H-region.The difference in the L-region may be due to the neglecting of someenergy terms present in the fundamental model in [24]. Because ofthe model simplification, the following terms are neglected: tem-perature change due to mass variations, temperature change dueto external stream, energy transfer due to diffusion, and tempera-ture change due to heat released by reactions. An improvement ofthe simplified physical model (1a) and (1b) can be made by addinga term that is proportional to the temperature in the equations. Fig.7 demonstrates the model errors obtained in the same way as de-scribed for Fig. 3 when adding a term �kfTf to the right hand side of(1a) and letting kf 2 [0,0.001]. This improvement obviously re-duces the difference between WA1 and the model error model esti-mates in the L-region.

In order to find a weight function WA(s) that covers the overallerrors of a nominal model to ensure controller robustness, we statethat:

10–3 10–2 10–1 100 10110–2

10–1

100

Am

plitu

de

10–3 10–2 10–1 100 10110–2

10–1

100

Frequency (rad/min)

Am

plitu

de

Fig. 7. Improvement of weight function WA1(s) (dot-dashed line) by adding �kfTf inthe model.

The amplitude of WA(s) should be greater than the upperbound of the 99% confidence interval of the model error modelestimate.

The upper plot in Fig. 8 illustrates the model error model (solidline) of the OC nominal model with N = 7, and a weight function(dashed line) that covers the overall errors.

In the H-region, the large amplitude values of the model errormodel estimate is most likely caused by the measurement noise,since the amplitude of the process must be small in this frequencyregion due to the process physics. It is unnecessary to take jWA(ix)jgreater than the 99% confidence interval of the model error modelestimate in the H-region, while it is still reasonable to have itgreater than jWA1(ix)j. The dashed weight function WA(s) shownin the upper plot of Fig. 8 is thus actually a conservative selection.

Based on the analysis presented above, a weight function WA(s)that covers the overall model errors of a nominal model is definedas follows:

In the L- and M-region, the amplitude of WA(s) is defined tobe greater than the upper bound of the 99% confidence intervalof the model error model estimate.

In the H-region, the amplitude of WA(s) is defined to begreater than WA1(ix).

An improved weight function is shown with the dot-dashed linein the upper plot of Fig. 8, and the function is

WAðsÞ ¼0:06ðsþ 0:0135Þðs2 þ 2sþ 0:3Þðsþ 0:81Þðs2 þ 0:0267sþ 0:002Þ ð26Þ

The lower plot of Fig. 8 illustrates weight functions selected forthe OC nominal models with N = 3, 5, 7, 17, respectively, which canbe used to analyze control system robustness. In general, it can beseen from the figure that higher order models result in smaller er-ror bounds.

10–3 10–2 10–1 10010

–3

Frequency (rad/min)

10–3

10–2

10–1

100

101

10–1

100

Am

plitu

de

N=3N=5N=7N=17

Frequency (rad/min)

Fig. 8. Upper: weight function (dashed line) of the overall errors defined based onthe estimate of the model error models of the OC nominal model with N = 7, and animproved weight function WA(s) (dot-dashed line); lower: weight functions of themodel errors of the OC nominal models in different orders.

Fig. 9. A closed-loop robust control of a perturbed process with additiveuncertainty.

L. Ding et al. / Journal of Process Control 19 (2009) 539–549 545

The discussion of the model errors estimated in different waysis helpful to get better understanding of the error sources, and inthe future this may help to improve a reduced model by minimiz-ing model errors caused by different reasons. For example, to min-imize the errors in M-region by improving the order reductionmethods, to minimize the errors in L-region by re-including someterms in the physical process model that were removed when sim-plifying it.

4. Application of the reduced models on process control andstate estimation

In this section, the reduced models are used for controller syn-thesis and design of a Kalman filter. The model quality and appli-cability can be evaluated by direct applications of the reducedmodels in process control and state estimation. The control robust-ness and the residuals of state estimation will be analyzed.

4.1. Process robust control

The estimated model error models provide a description of themodel errors in the frequency domain which is helpful for robustcontroller synthesis.

Fig. 9 illustrates a closed-loop control system for a perturbedprocess with an additive uncertainty, where DAWA = D and the per-turbed process bG þ D is the same as that represented by (25). Theweight function WA gives an upper bound to the additive uncer-tainty and DA represents a normalized dynamics that satisfieskDAk1 6 1.

Robust control synthesis based on the H1 loop shaping proce-dure computes a stabilizing H1 controller K for the plant bG toshape the singular value plot of the loop transfer function bGK.Roughly speaking, the function bGK is shaped to have desired loopshape Gd with accuracy c [25], then,

rðbGðjxÞKðjxÞÞP 1c

�rðGdðjxÞÞ for all x < x0 ð27Þ

�rðbGðjxÞKðjxÞÞ 6 crðGdðjxÞÞ for all x > x0 ð28Þ

where x0 is the 0 db crossover frequency of the singular value plotof Gd(jx).

The accuracy c is defined as [25,26]:

c ¼minK

I

K

� �I þ GKð Þ�1 GI½ �

��������1

ð29Þ

Robustness margin of a closed-loop system controlled by the ro-bust controller is defined as � = c�1. Obviously, a small value of cimplies that the controlled system has a large robustness margin[26].

Applying the robust control approach from [17,16,26], a closed-loop control system of a perturbed process can be illustrated usinga general control configuration consisting of a generalized plant, acontroller K and a normalized uncertainty, in our case DA. From Fig.9, the generalized plant is obtained as

ð30Þ

where uD = (DA) yD, and assuming r = 0 then e = �y and u = �Ky.An ND structure [17,16,26] of the closed-loop system is then gi-

ven as

yD

z

� �¼

N11 N12

N21 N22

� �uD

w

� �ð31Þ

where N11 = N12 = �WAKS, N21 = N22 = WpS, and S ¼ ðI þ bGKÞ�1 is thesystem sensitivity function.

Using the ND structure, the following criterion [17] must holdfor robust stability of the closed-loop system:

Assume that the system is nominally stable, DA is a set of sta-ble full complex matrices, and kDAk1 6 1, then the perturbedsystem is robustly stable if and only if

�rðN11ðixÞÞ < 1 8x ð32Þ

The closed-loop system illustrated by Fig. 9 is thus robustly sta-ble if and only if

�rðKðixÞSðixÞÞ < jWAðixÞj�1 8 x ð33Þ

or equivalently

�rðTðixÞÞ < jbGðixÞjjWAðixÞj�1 8 x ð34Þ

where TðixÞ ¼ bGðixÞKðixÞðI þ bGðixÞKðixÞÞ�1.The robust stability criterion (34) will be utilized for an applica-

tion example of robust process control discussed in the nextsubsection.

4.2. Digester temperature control based on the H1 loop shapingprocedure

For process control one prefers to use a low-order model. In thisstudy we choose nominal models with different N in control syn-thesis for comparing the system robust stability and performance.

In order to achieve good disturbance rejection and commandtracking, we want to synthesize a controller so that the open-loopfunction of the controlled system has large amplitude in low fre-quencies. For robust stability, we want the open-loop function tohave small magnitude in high frequencies. In general, it is reason-able that the function has a slope of �1 in the crossover region andat least a roll-off of 2. Based on a great number of simulation exper-iments we select an open-loop function as follows:

GdðsÞ ¼0:03

sðs=0:04þ 1Þðs=1:1þ 1Þ ð35Þ

A process control with robust stability can be achieved by applyingthe H1 loop shaping procedure based on the nominal models withdifferent choice of N.

Fig. 10 demonstrates the desired open-loop function Gd and theopen-loop function bGK where K is a synthesized controller basedon a nominal model with N = 3. The loop shaping procedure guar-antees a bGK satisfying (27) and (28).

The left plot in Fig. 11 illustrates the robust stability of the con-troller synthesized using the desired open-loop function (35), andit shows that the robust stability criterion (34) holds. The weightfunction plotted with a solid line in the lower plot of Fig. 8 is usedfor robustness analysis.

To have some appreciation for how the controller would behavewith the real process, it is tested using the frequency domain solu-tion of the infinite dimensional model (1a) and (1b) as the process,

102

100

–100

–50

0

(rad/sec)

Sin

gula

r V

alue

s (d

b)

σ(Gd)γ

σ(Gd)/γ

σ(L)σ(G

d)

Frequency (rad/min)

Fig. 10. Robust control synthesized using the loop shaping procedure based on anominal model with N = 3.

546 L. Ding et al. / Journal of Process Control 19 (2009) 539–549

i.e. using the infinite dimensional transfer function (5) and (6),where the model is perturbed by taking different parameter valuesfrom the range shown in Table 1. In the rest of this section, all thesimulations of the perturbed process use the infinite dimensionaltransfer function as the simulation model. The upper plot in Fig.12 demonstrates the nominal time response (solid line) and theperturbed time responses (dotted lines) of the controlled system,where there exist obvious oscillations caused by the parameterperturbation.

Next we test another controller synthesized using the followingdesired open-loop transfer function:

GdðsÞ ¼0:03

sðs=0:02þ 1Þðs=1:1þ 1Þ ð36Þ

where a greater dominating time constant in Gd will make the ref-erence tracking slightly slower but also allow increased stabilitymargins. The right plot of Fig. 11 shows that the system is more ro-bustly stable than the left plot, and the lower plot of Fig. 12 demon-strates better performance when subject to perturbations. Thenominal performance is degraded in the sense of a slower timeresponse.

A higher order nominal model usually has better accuracy thana lower order nominal model as can be seen in Fig. 8. Using a high-er order model may help the robust control synthesis to achieveboth high performance and good robustness. Thus we test an OCnominal model with N = 7 in the controller synthesis, and selectthe open-loop function (35) in order to achieve a faster time re-sponse. The weight function for this nominal model is plottedusing dot-dashed line in Fig. 8, and robustness analysis shows thatthe system is robustly stable. The upper plot in Fig. 13 shows per-turbed step responses and illustrates the better performance com-pared to that obtained using the OC model with N = 3.

10–2 100–120

–100

–80

–60

–40

–20

0

20

Sin

gula

r V

alue

s (d

b)

σ(T)σ(G/W

A)

Frequency (rad/min)

Fig. 11. Robust stability of a closed-loop system according to criterion (34). Controllertransfer function (35); right: choosing open-loop transfer function (36).

Now we test a controller synthesized based on the 17th ordernominal model and compare the system performance with thatusing lower order models. The weight function defined for thishigh order model is shown in Fig. 8 using a dotted line, and the de-sired open-loop transfer function (35) is used. The lower plot in Fig.13 illustrates the step response of the perturbed system, and thereis no obvious difference between this plot and the upper one thatshows the control system synthesized based on the nominal modelwith N = 7.

In conclusion, using the nominal model with N = 3, it is difficultto achieve good performance while tolerating parameter perturba-tions. If a slightly degraded performance can be accepted then theparameter perturbations are tolerated much better. Using a nomi-nal model with a greater N, e.g. N = 7, the system performance androbustness are obviously improved. However, the grade ofimprovement is not proportional to the increase of model order,and this can be seen when the nominal model with N = 17 is cho-sen. Thus it may be unnecessary to choose a model with very highorder. The low-order nominal models have advantages in control-ler implementation and computational time, and it is possible toselect an appropriate low-order model by making a trade-off be-tween simplicity of the model and robustness/performance of theclosed-loop system.

4.3. Application of the reduced models in state estimation

A major motivation for using a physical process model is thatthe state x then has a physical interpretation. Thus an estimate xof the state can provide valuable information about the current sta-tus of the process, in particular of quantities that cannot be mea-sured directly.

The standard method of state estimation utilizes a stateobserver

_x ¼ Axþ Buþ Kf ðy� Cx� DuÞ ð37Þ

which requires a low-order process model of the form

_x ¼ Axþ BuþMw1

y ¼ Cxþ Duþw2

where w1 and w2 are process disturbance and measurement distur-bance, respectively. In order to show the potential of the reducedorder model (10) for this purpose, a state observer is designed un-der the assumption that w1 and w2 are uncorrelated white noise sig-nals with variance R1 and R2, respectively. For simplicity, it is alsoassumed that w1 consists of 2(N + 1) uncorrelated white noise sig-nals with variance r1, each affecting one state variable. ThusR1 = r1I2(N+1) and M = I2(N+1). An identified process model with

10–2 100–120

–100

–80

–60

–40

–20

0

20

Sin

gula

r V

alue

s (d

b)

σ(T)σ(G/W

A)

Frequency (rad/min)

s are synthesized based on a nominal model with N = 3. Left: choosing open-loop

0 500 1000 1500 2000–1.5

–1

–0.5

0

0.5

1

Out

put

0 500 1000 1500 2000–0.2

–0.1

0

0.1

0.2

Time (min)

Res

idua

l

Fig. 14. Upper: Estimated (solid) and measured (dashed) process output. Lower:Observer residual.

0 100 200 300 400 500–0.5

0

0.5

1

1.5

Time (min)

Am

plitu

de

0 100 200 300 400 500–0.5

0

0.5

1

1.5

Time (min)

Am

plitu

de

Fig. 12. Step responses of the nominal system (solid line) and the perturbed infinitedimensional system (dotted line). The controllers are synthesized based on a no-minal model with N = 3. Upper: choosing open-loop transfer function (35); lower:choosing open-loop transfer function (36).

0 100 200 300 400 500–0.5

0

0.5

1

1.5

Time (min)

Am

plitu

de

0 100 200 300 400 500–0.5

0

0.5

1

1.5

Time (min)A

mpl

itude

Fig. 13. Step responses of the nominal system (solid line) and the perturbed infinitedimensional system (dotted line). The desired open-loop transfer function is chosenas (35). Upper: the controller is synthesized based on a nominal model with N = 7;lower: the controller is synthesized based on a nominal model with N = 17.

L. Ding et al. / Journal of Process Control 19 (2009) 539–549 547

N = 7 is used in the experiments. A Kalman filter, i.e. (37) with opti-mal value of the observer feedback gain Kf is then obtained by solv-ing a certain Riccati equation [17].

The Kalman filter is designed by manually adjusting the designparameters r1 and R2 to make the observer residual r ¼ y� Cx� Duresemble white noise for a set of measurement data of input u andoutput y. The result is shown in Fig. 14.

In order to determine the source of the estimation error, the ob-server is also applied to simulated output data y, generated usingthe frequency domain solution of (1), i.e. the frequency responseTf(z, ix) for z = 1. An estimated observer output y is then generatedusing the same observer gain Kf as above and the result is shown inFig. 15. The model reduction is thus the only source of observerresidual in this case and, since it is significantly smaller comparedto Fig. 14, it is concluded that the major source of estimation errorcomes from imperfections in the process model (1), not the modelreduction.

The state vector of the low-order model represents the temper-atures in the collocation points z = z0,z1, . . . ,zN+1. To obtain the tem-peratures in an arbitrary point z one may use (13) and (14) toobtain the interpolation

T f ðz; tÞ ¼ Czxf ðtÞ þ DzuðtÞ ð38ÞTcðz; tÞ ¼ CzxcðtÞ þ DzuðtÞ ð39Þ

where Cz = [l1(z) l2(z) � � � lN+1(z)] and Dz = l0(z). The state Tf at two dif-ferent positions (z = 0.42 and z = 1), estimated using a process out-put generated by the frequency domain solution of (1) are shown inFig. 16 (upper). It is compared to the process state calculated usingthe frequency domain solution of (1), i.e. the frequency responseTf(z, ix). The agreement between estimated and calculated state

shows that an observer for the reduced order process is capableof estimating the state of the PDE model (1).

To show that the observer can also estimate the state of the ac-tual process, the Tf temperature at position z = 0.42 is estimatedusing measurement data of the output. The estimate is comparedto a measurement of the temperature at this position, denoted c6in Fig. 1. The result is displayed in Fig. 16 (upper) and it is clear thatthey agree quite well. For reference, the simulated process state atthe same position, calculated using the frequency domain solutionof (1), is shown in the same plot.

0 500 1000 1500 2000–1.5

–1

–0.5

0

0.5

1

Out

put

0 500 1000 1500 2000–0.2

–0.1

0

0.1

0.2

Time (min)

Res

idua

l

Fig. 15. Upper: Estimated process output (solid) and output generated from thefrequency domain solution of (1) (dashed). Lower: Observer residual.

500 550 600 650 700 750 800 850 900–1.5

–1

–0.5

0

Pro

cess

sta

te

500 550 600 650 700 750 800 850 900–1.5

–1

–0.5

0

Time (min)

Pro

cess

sta

te

Fig. 16. Upper: Process state Tf at position z = 0.42 and z = 0.7 estimated using theoutput calculated from the frequency domain solution of (1) and compared to pr-ocess state calculated from the frequency domain solution of (1) (dotted). Lower:Process state Tf at the c6 position (z = 0.42), estimated (solid) using measured pr-ocess output and compared to measured c6 temperature (dashed) and simulatedstate (dotted) calculated from the frequency domain solution of (1).

548 L. Ding et al. / Journal of Process Control 19 (2009) 539–549

5. Conclusions

The problem of applying DPS models for robust control andstate estimation is addressed. By integrating model reduction withparameter identification and model uncertainty analysis, it is dem-onstrated how an appropriate trade-off between complexity androbust performance can be achieved. Measurement data from acontinuous paper pulp digester is used throughout the study.

Low-order model approximation obtained using the OC ap-proach is proposed to solve the problems due to the infinite dimen-sionality. The unknown parameters of the process model areidentified based on the reduced models of different orders andthe parameter estimates are shown to converge to small intervalswith increasing model order.

The errors of the reduced models are estimated and analyzedusing two different approaches. Comparing a low-order model tothe infinite dimensional model with different parameter values ex-

poses the errors mainly caused by parameter perturbation andmodel reduction. Estimating a model error model provides a pic-ture of the overall errors including those caused by other reasons,such as model simplifications. By combining the results from thetwo approaches, a model error weight function used for robustcontrol can be set up.

The loop shaping procedure is applied to synthesize H1 optimalcontrollers. Simulation experiments demonstrate that it is possibleto select an appropriate low-order model for controller synthesisby taking a trade-off between simplicity of the model and robust-ness/performance of the closed-loop system.

A Kalman filter is designed using a reduced process model in or-der to produce estimates of the process state. By comparing withmeasurement data it is demonstrated that the Kalman filter pro-vides reliable estimates of the state at an arbitrary spatial point.Such an observer may be used for process monitoring and faultdetection.

Acknowledgements

The authors would like to thank the Hjalmar Lundbohm ResearchCenter for financial support. Measurement data from the Husumpaper plant provided by M-real are greatly appreciated.

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